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#### A Review of Some Results of Kac’s Program in Kinetic Theory

, M.Sc. Thesis Sharif University of Technology ; Safdari, Mohammad (Supervisor)
Abstract

This thesis is devoted to the study of suitable functional frameworks for proving propagation of chaos and mean-field limit of nonlinear Vlasov-type equations for indistinguishable particles. The Kac model is a classic example and the motivation of this functional structure. On the one hand, Kac considered a spatially homogeneous gas, and on the other hand, he reduced the problem to one-dimensional collisions and dropped the conservation of momentum assumption. Therefore, he obtained a simplified version of the Boltzmann equation, which is named after him, the Kac equation. The main results which we study are quantitative estimates

on the decay of fluctuations around the deterministic...

on the decay of fluctuations around the deterministic...

#### Shear Viscosities of a Hot Fermi Gas in the Presence of a Strong Magnetic Field

, M.Sc. Thesis Sharif University of Technology ; Sadooghi, Neda (Supervisor)
Abstract

Relativistic hydrodynamics and kinetic theory are often used to analyze the experimental data from heavy ion collisions. One of the most important observations is the anisotropy in the number density of the detected particles after the collision in the longitudinal and perpendicular directions with respect to the reaction plane. On the other hand, very strong magnetic fields are created during the early stages of non-central heavy ion collisions. The presence of constant background magnetic fields, aligned in the perpendicular direction to the reaction plane, can be one of reasons for the aforementioned anisotropy in the number density of detected particles. In this thesis, we study the...

#### Existence and Regularity of Renormalized Solutions for Boltzmann Equation

, M.Sc. Thesis Sharif University of Technology ; Fotouhi, Morteza (Supervisor)
Abstract

In 1989 DiPerna and Lions proved the stability of Boltzmann’s evolution equation using their theory of renormalized solutions. Their result , for the first time, proved the existence of solutions without extra assumptions on the initial condition and on arbitrary time intervals. While the wellknown Grad’s cut-off assumption is present in the original theory, there have been successful generalizations to account for the singular case. Also the renormalization theory has shed light on limiting regimes of the Boltzmann equation. Here we discuss the new techniques that are essential for such generalizations

#### Development of Spectral Difference Lattice Boltzmann Method for Solution of Compressible Flows

, Ph.D. Dissertation Sharif University of Technology ; Hejranfar, Kazem (Supervisor)
Abstract

In this research, the spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional (2D) inviscid and viscous compressible flows on the structured and unstructured meshes. The compressible form of the discrete Boltzmann-BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is used. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution functions are stored at the solution points while the fluxes are calculated...

#### Development of Compact Finite Difference Boltzmann Method for Simulating Compressible Rarefied Gas Flow

, M.Sc. Thesis Sharif University of Technology ; Hejranfar, Kazem (Supervisor) ; Fouladi, Nematollah (Co-Supervisor)
Abstract

In this work, a high-order accurate gas kinetic scheme based on the compact finite-difference Boltzmann method (CFDBM) is developed and applied for simulating the compressible rarefied gas flows. Here, the Shakhov model of the Boltzmann equation is considered and the spatial derivative term in the resulting equation is discretized by using the fourth-order compact finite-difference method and the time integration is performed by using the third-order TVD Runge-Kutta method. A filtering procedure with three discontinuity-detecting sensors is applied and examined for the stabilization of the solution method especially for the problems involving the discontinuity regions such as the shock. The...